Integrand size = 24, antiderivative size = 37 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )}{\alpha } \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {738, 210} \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]
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Rule 210
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-4 \alpha ^2-r^2} \, dr,r,\frac {-2 \alpha ^2-2 k r}{\sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )\right ) \\ & = -\frac {\arctan \left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )}{\alpha } \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {2 \arctan \left (\frac {\sqrt {2} \sqrt {h} r-\sqrt {-\alpha ^2-2 k r+2 h r^2}}{\alpha }\right )}{\alpha } \]
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Time = 0.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 k r +2 \sqrt {-\alpha ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}}}\) | \(52\) |
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (\frac {\sqrt {2 \, h r^{2} - \alpha ^{2} - 2 \, k r} {\left (\alpha ^{2} + k r\right )}}{2 \, \alpha h r^{2} - \alpha ^{3} - 2 \, \alpha k r}\right )}{\alpha } \]
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\[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=\int \frac {1}{r \sqrt {- \alpha ^{2} + 2 h r^{2} - 2 k r}}\, dr \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {\arcsin \left (\frac {k}{\sqrt {2 \, \alpha ^{2} h + k^{2}}} + \frac {\alpha ^{2}}{\sqrt {2 \, \alpha ^{2} h + k^{2}} r}\right )}{\alpha } \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=\frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {h} r - \sqrt {2 \, h r^{2} - \alpha ^{2} - 2 \, k r}}{\alpha }\right )}{\alpha } \]
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Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr=-\frac {\ln \left (\frac {\sqrt {-\alpha ^2}\,\sqrt {-\alpha ^2+2\,h\,r^2-2\,k\,r}}{r}-\frac {\alpha ^2}{r}-k\right )}{\sqrt {-\alpha ^2}} \]
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